rethinking package (McElreath 2020)
“Automated” packages: rstanarm and brms
posterior, tidybayes, bayesplot)lm() and summary()Likelihood:
\[y \sim Normal(\mu, \sigma)\]
Priors:
\[\mu \sim Normal(45, 10)\]
\[\sigma \sim HalfNormal(0, 3)\]
Skip prior predictive simulation (for now)
ulam() modelNote the correspondence to the model statements.
50% of iterations are used for warm-up (default)
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Shown by Vehtari et al. (2021) to be better than trace plots.
ulam objectsInference for Stan model: ulam_cmdstanr_36d52c78ab5e039bffe696e0e0489438-202302200927-1-8b32ac.
4 chains, each with iter=10000; warmup=5000; thin=1;
post-warmup draws per chain=5000, total post-warmup draws=20000.
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
mu 48.53 0.01 0.71 47.12 48.10 48.53 48.96 49.95 8300 1
sigma 1.91 0.01 0.59 1.12 1.50 1.79 2.18 3.40 7913 1
lp__ -8.22 0.01 1.11 -11.20 -8.66 -7.88 -7.43 -7.13 5780 1
Samples were drawn using NUTS(diag_e) at Mon Feb 20 09:27:21 2023.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at
convergence, Rhat=1).
rethinking::precis(): shorter summaryn_effn_eff is small
Rhat4When sampling fails, it usually fails very badly and very obviously.
Gelman’s Folk Theorem of Statistical Computing:
When you have computational problems, often there’s a problem with your model.
ulam object contains samples:
mu sigma
1 47.5197 1.32060
2 50.3552 3.12731
3 47.8664 1.43598
4 48.1501 2.05984
5 48.9498 1.56121
6 47.5633 1.85286
89% percentile interval:
89% HDPI:
(Usually nearly) identical for symmetrical distributions:
(Can) differ for asymmetrical distributions:
rethinking::HPDI()HDInterval::hdi()coda::HPDinterval()